Optimal. Leaf size=309 \[ -\frac {(b c-a d) x \sqrt {a-b x^4}}{4 c d \left (c-d x^4\right )}+\frac {\sqrt [4]{a} b^{3/4} (3 b c+a d) \sqrt {1-\frac {b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c d^2 \sqrt {a-b x^4}}-\frac {3 \sqrt [4]{a} (b c-a d) (b c+a d) \sqrt {1-\frac {b x^4}{a}} \Pi \left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 d^2 \sqrt {a-b x^4}}-\frac {3 \sqrt [4]{a} (b c-a d) (b c+a d) \sqrt {1-\frac {b x^4}{a}} \Pi \left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 d^2 \sqrt {a-b x^4}} \]
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Rubi [A]
time = 0.17, antiderivative size = 309, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {424, 537, 230,
227, 418, 1233, 1232} \begin {gather*} \frac {\sqrt [4]{a} b^{3/4} \sqrt {1-\frac {b x^4}{a}} (a d+3 b c) F\left (\left .\text {ArcSin}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c d^2 \sqrt {a-b x^4}}-\frac {3 \sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} (b c-a d) (a d+b c) \Pi \left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}};\left .\text {ArcSin}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 d^2 \sqrt {a-b x^4}}-\frac {3 \sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} (b c-a d) (a d+b c) \Pi \left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}};\left .\text {ArcSin}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 d^2 \sqrt {a-b x^4}}-\frac {x \sqrt {a-b x^4} (b c-a d)}{4 c d \left (c-d x^4\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 227
Rule 230
Rule 418
Rule 424
Rule 537
Rule 1232
Rule 1233
Rubi steps
\begin {align*} \int \frac {\left (a-b x^4\right )^{3/2}}{\left (c-d x^4\right )^2} \, dx &=-\frac {(b c-a d) x \sqrt {a-b x^4}}{4 c d \left (c-d x^4\right )}-\frac {\int \frac {-a (b c+3 a d)+b (3 b c+a d) x^4}{\sqrt {a-b x^4} \left (c-d x^4\right )} \, dx}{4 c d}\\ &=-\frac {(b c-a d) x \sqrt {a-b x^4}}{4 c d \left (c-d x^4\right )}+\frac {\left (3 \left (a^2-\frac {b^2 c^2}{d^2}\right )\right ) \int \frac {1}{\sqrt {a-b x^4} \left (c-d x^4\right )} \, dx}{4 c}+\frac {(b (3 b c+a d)) \int \frac {1}{\sqrt {a-b x^4}} \, dx}{4 c d^2}\\ &=-\frac {(b c-a d) x \sqrt {a-b x^4}}{4 c d \left (c-d x^4\right )}+\frac {\left (3 \left (a^2-\frac {b^2 c^2}{d^2}\right )\right ) \int \frac {1}{\left (1-\frac {\sqrt {d} x^2}{\sqrt {c}}\right ) \sqrt {a-b x^4}} \, dx}{8 c^2}+\frac {\left (3 \left (a^2-\frac {b^2 c^2}{d^2}\right )\right ) \int \frac {1}{\left (1+\frac {\sqrt {d} x^2}{\sqrt {c}}\right ) \sqrt {a-b x^4}} \, dx}{8 c^2}+\frac {\left (b (3 b c+a d) \sqrt {1-\frac {b x^4}{a}}\right ) \int \frac {1}{\sqrt {1-\frac {b x^4}{a}}} \, dx}{4 c d^2 \sqrt {a-b x^4}}\\ &=-\frac {(b c-a d) x \sqrt {a-b x^4}}{4 c d \left (c-d x^4\right )}+\frac {\sqrt [4]{a} b^{3/4} (3 b c+a d) \sqrt {1-\frac {b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c d^2 \sqrt {a-b x^4}}+\frac {\left (3 \left (a^2-\frac {b^2 c^2}{d^2}\right ) \sqrt {1-\frac {b x^4}{a}}\right ) \int \frac {1}{\left (1-\frac {\sqrt {d} x^2}{\sqrt {c}}\right ) \sqrt {1-\frac {b x^4}{a}}} \, dx}{8 c^2 \sqrt {a-b x^4}}+\frac {\left (3 \left (a^2-\frac {b^2 c^2}{d^2}\right ) \sqrt {1-\frac {b x^4}{a}}\right ) \int \frac {1}{\left (1+\frac {\sqrt {d} x^2}{\sqrt {c}}\right ) \sqrt {1-\frac {b x^4}{a}}} \, dx}{8 c^2 \sqrt {a-b x^4}}\\ &=-\frac {(b c-a d) x \sqrt {a-b x^4}}{4 c d \left (c-d x^4\right )}+\frac {\sqrt [4]{a} b^{3/4} (3 b c+a d) \sqrt {1-\frac {b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c d^2 \sqrt {a-b x^4}}+\frac {3 \sqrt [4]{a} \left (a^2-\frac {b^2 c^2}{d^2}\right ) \sqrt {1-\frac {b x^4}{a}} \Pi \left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 \sqrt {a-b x^4}}+\frac {3 \sqrt [4]{a} \left (a^2-\frac {b^2 c^2}{d^2}\right ) \sqrt {1-\frac {b x^4}{a}} \Pi \left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 \sqrt {a-b x^4}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 4 in
optimal.
time = 10.23, size = 342, normalized size = 1.11 \begin {gather*} \frac {x \left (-b (3 b c+a d) x^4 \sqrt {1-\frac {b x^4}{a}} \left (-c+d x^4\right ) F_1\left (\frac {5}{4};\frac {1}{2},1;\frac {9}{4};\frac {b x^4}{a},\frac {d x^4}{c}\right )+\frac {5 c \left (-5 a c \left (4 a^2 d+b^2 c x^4-a b d x^4\right ) F_1\left (\frac {1}{4};\frac {1}{2},1;\frac {5}{4};\frac {b x^4}{a},\frac {d x^4}{c}\right )-2 (-b c+a d) x^4 \left (a-b x^4\right ) \left (2 a d F_1\left (\frac {5}{4};\frac {1}{2},2;\frac {9}{4};\frac {b x^4}{a},\frac {d x^4}{c}\right )+b c F_1\left (\frac {5}{4};\frac {3}{2},1;\frac {9}{4};\frac {b x^4}{a},\frac {d x^4}{c}\right )\right )\right )}{5 a c F_1\left (\frac {1}{4};\frac {1}{2},1;\frac {5}{4};\frac {b x^4}{a},\frac {d x^4}{c}\right )+2 x^4 \left (2 a d F_1\left (\frac {5}{4};\frac {1}{2},2;\frac {9}{4};\frac {b x^4}{a},\frac {d x^4}{c}\right )+b c F_1\left (\frac {5}{4};\frac {3}{2},1;\frac {9}{4};\frac {b x^4}{a},\frac {d x^4}{c}\right )\right )}\right )}{20 c^2 d \sqrt {a-b x^4} \left (-c+d x^4\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.27, size = 328, normalized size = 1.06
method | result | size |
default | \(\frac {\left (a d -b c \right ) x \sqrt {-b \,x^{4}+a}}{4 c d \left (-d \,x^{4}+c \right )}+\frac {\left (\frac {b^{2}}{d^{2}}+\frac {b \left (a d -b c \right )}{4 d^{2} c}\right ) \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}-\frac {3 \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (d \,\textit {\_Z}^{4}-c \right )}{\sum }\frac {\left (a^{2} d^{2}-b^{2} c^{2}\right ) \left (-\frac {\arctanh \left (\frac {-2 b \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 a}{2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {-b \,x^{4}+a}}\right )}{\sqrt {\frac {a d -b c}{d}}}-\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} d \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \EllipticPi \left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, \frac {\sqrt {a}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d}{\sqrt {b}\, c}, \frac {\sqrt {-\frac {\sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, c \sqrt {-b \,x^{4}+a}}\right )}{\underline {\hspace {1.25 ex}}\alpha ^{3}}\right )}{32 c \,d^{3}}\) | \(328\) |
elliptic | \(\frac {\left (a d -b c \right ) x \sqrt {-b \,x^{4}+a}}{4 c d \left (-d \,x^{4}+c \right )}+\frac {\left (\frac {b^{2}}{d^{2}}+\frac {b \left (a d -b c \right )}{4 d^{2} c}\right ) \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}-\frac {3 \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (d \,\textit {\_Z}^{4}-c \right )}{\sum }\frac {\left (a^{2} d^{2}-b^{2} c^{2}\right ) \left (-\frac {\arctanh \left (\frac {-2 b \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 a}{2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {-b \,x^{4}+a}}\right )}{\sqrt {\frac {a d -b c}{d}}}-\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} d \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \EllipticPi \left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, \frac {\sqrt {a}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d}{\sqrt {b}\, c}, \frac {\sqrt {-\frac {\sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, c \sqrt {-b \,x^{4}+a}}\right )}{\underline {\hspace {1.25 ex}}\alpha ^{3}}\right )}{32 c \,d^{3}}\) | \(328\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a - b x^{4}\right )^{\frac {3}{2}}}{\left (- c + d x^{4}\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a-b\,x^4\right )}^{3/2}}{{\left (c-d\,x^4\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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